Aalborg Universitet

Containment and Consensus-Based Distributed Coordination Control to AchieveBounded Voltage and Precise Reactive Power Sharing in Islanded AC Microgrids

Han, Renke; Meng, Lexuan; Ferrari-Trecate, Giancarlo; Coelho, Ernane Antônio Alves;Quintero, Juan Carlos Vasquez; Guerrero, Josep M.Published in:I E E E Transactions on Industry Applications

DOI (link to publication from Publisher):10.1109/TIA.2017.2733457

Publication date:2017

Document VersionVersion created as part of publication process; publisher's layout; not normally made publicly available

Link to publication from Aalborg University

Citation for published version (APA):Han, R., Meng, L., Ferrari-Trecate, G., Coelho, E. A. A., Quintero, J. C. V., & Guerrero, J. M. (2017).Containment and Consensus-Based Distributed Coordination Control to Achieve Bounded Voltage and PreciseReactive Power Sharing in Islanded AC Microgrids. I E E E Transactions on Industry Applications, 53(6), 5187-5199. https://doi.org/10.1109/TIA.2017.2733457

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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 53, NO. 6, NOVEMBER/DECEMBER 2017 5187

Containment and Consensus-Based DistributedCoordination Control to Achieve Bounded Voltage

and Precise Reactive Power Sharing inIslanded AC Microgrids

Renke Han , Student Member, IEEE, Lexuan Meng , Member, IEEE,Giancarlo Ferrari-Trecate, Senior Member, IEEE, Ernane Antonio Alves Coelho, Member, IEEE,

Juan C. Vasquez, Senior Member, IEEE, and Josep M. Guerrero , Fellow, IEEE

Abstract—This paper presents a novel distributed approach toachieve both bounded voltage and accurate reactive power sharingregulation in ac microgrid. The coupling/trade-off effects betweenbus voltages and reactive power sharing regulation are first ana-lyzed in detail to provide a guideline for coordinated control de-sign. Furthermore, a containment and consensus-based distributedcoordination controller is proposed, by which the bus voltage mag-nitudes can be bounded within a reasonable range, instead ofonly controlling average voltage value. Furthermore, the accu-rate reactive power sharing between distributed generators canbe achieved simultaneously. Then, a detailed small-signal model isdeveloped to analyze the stability of the system and the sensitivityof different parameters. Experimental results are presented andcompared, where the controller performance, robust performanceunder communication failure, and plug-and-play operation aresuccessfully verified.

Index Terms—Bounded voltage, consensus algorithm, contain-ment algorithm, coordinated control, microgrid (MG), reactivepower sharing.

I. INTRODUCTION

THE MICROGRID (MG) concept provides a promisingmean of integrating large amount of distributed generators

(DGs) into the power grid and improving reliability [1]. Basedon the MG concept, hierarchical control architecture, which was

Manuscript received April 28, 2017; revised June 21, 2017; accepted July 24,2017. Date of publication July 27, 2017; date of current version November 20,2017. Paper 2017-PSEC-0368.R1, presented at the 2017 IEEE Applied PowerElectronics Specialists Conference, Tampa, FL, USA, Mar. 26–30, and approvedfor publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS bythe Power System Engineering Committee of the IEEE Industry ApplicationsSociety. (Corresponding author: Renke Han.)

R. Han, L. Meng, J. C. Vasquez, and J. M. Guerrero are with the EnergyTechnology Department, Aalborg University, Aalborg 9220, Denmark (e-mail:rha@et.aau.dk; lme@et.aau.dk; juq@et.aau.dk; joz@et.aau.dk).

G. Ferrari-Trecate is with the Laboratoire d’Automatique, Ecole Polytech-nique Federale de Lausanne, Lausanne 1015, Switzerland (e-mail: giancarlo.ferraritrecate@epfl.ch).

E. A. A. Coelho is with the Faculdade de Engenharia Eletrica, UniversidadeFederal de Uberlandia, Faculdade de Engenharia Eletrica, Uberlandia 38400-902, Brazil (e-mail: ernane@ufu.br).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIA.2017.2733457

commonly applied in power systems, has been adopted and mod-ified to coordinate different control objectives and to standardizethe MG operation [2], [3]. Typically, the primary level deals withthe voltage/current regulation and power sharing local control.The secondary level is used to restore the system frequency andvoltage to nominal values, regardless of load changes. The ter-tiary level deals with the energy management and optimizationissues.

For the primary and secondary levels, one of main challengesis to achieve the coordination between reactive power sharingand output voltage magnitudes regulation. The reactive power-to-voltage (Q–V) droop control [4] is applied to achieve reactivepower sharing in a decentralized manner. However, the Q–Vdroop control is very sensitive to line impedance differencesthus producing inaccurate reactive power sharing and deviat-ing the voltage excessively when researchers try to correct itby increasing the droop coefficient value [5]. In [6], the reac-tive power sharing problem is analyzed and a two-step estima-tion method is proposed to calculate the local reactive load,based on which the droop gain is adjusted accordingly to im-prove the reactive power sharing. Both the effects from localloads and line impedances are considered to improve the powersharing performance. However, the load information should beidentified in advance. Then, with a centralized energy man-agement system, a dynamic virtual impedance is proposed inorder to satisfy the reactive power sharing requirements basedon the different load conditions [7]. However, the load informa-tion is also required by the centralized controller, which limitsthe expandability of the proposed controller. In addition, thecentralized controller can cause the single point of failure. In[8], according to the electrical topology of the MG, the relation-ship between active/reactive load changes and reactive powersharing error is analyzed, and a genetic algorithm is applied tooptimize the virtual impedance, in order achieve reactive powersharing. However, when the topology of MG is changed, thewhole optimization problem needs to be reformulated. Noticethat all the aforementioned controllers are focusing just on thereactive power sharing, while the problem of voltage recoveryis not considered at the same time.

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5188 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 53, NO. 6, NOVEMBER/DECEMBER 2017

Recently, distributed control algorithms [9]–[13] are comingup to stage finding their applications in the MG area [14]–[21].A distributed method is proposed in [14] to achieve reactivepower sharing through acquiring the average value of reactivepower. Similarly, in [15], considering the converters parametersrelated to the power limit and maximum active power capability,a reactive power sharing algorithm is developed. However, forthe above-mentioned two kinds of controllers, each distributedcontroller needs to know the output reactive power and outputvoltage magnitudes of all the other DGs in the MG. Thus, if oneDG fails, the operating DG number should be updated for allthe other controllers, which limits the flexibility and robustnessof the system. Furthermore, in [15], the voltage regulation is notconsidered, which may result in considerable voltage deviations.On the other hand, based on the distributed leader-followingtracking algorithm [9], another work [16] proposes a secondaryvoltage tracking strategy by using the feedback linearizationmethod, achieving the output voltage magnitudes tracking theleader. However, the problem of reactive power sharing is notconsidered during the controller design process. In another work[17], in order to evaluate the issues between accurate reactivepower sharing and voltage magnitude regulation, a simple trade-off analysis is proposed for the two control objectives withinthe secondary control level. Then, an averaging-based method[13] has been applied to achieve reactive power sharing andto keep the average voltage value equal to the nominal value.However, in this work only the average value of all output volt-age magnitudes is regulated, being possible large local voltagedeviations.

Alternatively, in [19], a droop-free distributed method is pro-posed to achieve power sharing and to fix the average outputvoltage to the nominal value. However, the system cannot beoperated in practical applications without the use of droop con-trol when the communication system is disabled or presents aglobal failure. Furthermore, only controlling the average voltagevalue is not enough for some operation conditions. For example,when one or several DGs are disconnected from the MG or thedifference among output line impedances is large, the voltagemagnitudes at some buses deviate out from the allowed limit,which can affect the power quality of the system, even thoughthe average voltage value is kept at the nominal value [22].Compared with the voltage deviation standard for electrical dis-tribution networks, e.g., IEEE 1547, the voltage deviations in anislanded MG should be smaller to guarantee stable of the MG.Accordingly, the existing literature only focus on regulating theaverage value of voltage magnitudes rather than bounding allbus voltage magnitudes in a reasonable range.

Thus, instead of only controlling the average voltage value,a more flexible control strategy is required to bound all busvoltage magnitudes into a reasonable range, and to achieve ac-curate reactive power simultaneously. Under more serious con-ditions, to guarantee the bus voltages are bounded and to providevoltage support for the system stability, the performance of re-active power sharing should be compromised to some extent.In addition, the coupling and trade-off effects between criticalsystem parameters, including droop gains, line impedances,voltage magnitude deviations, and reactive power sharing

Fig. 1. Simplified model of the microgrid.

errors should be analyzed in detail by considering differentconditions.

To solve the above-mentioned critical issues, a containment-based controller [23] is identified and considered as a reasonableand flexible approach. It can bound objects within a convexrange, while maintaining the distributed fashion, which makesit highly suitable for DG-based MG applications.

In this paper, there are four main technical contributions.First, coupling and trade-off effects between different controlparameters and objectives are analyzed within primary and sec-ondary control level. Second, a fully distributed coordinationcontrol scheme including containment-based and consensus-based algorithms is proposed realizing a well coordination be-tween bounded bus voltage and accurate reactive power sharing.Third, the small-signal model is derived for system stabilityanalysis and control parameter design. Finally, experimentalresults including control and robust performance comparison,especially under communication links failure and plug-and-playconditions, are shown to prove high resiliency of the proposedcontroller.

The paper is organized as follows. In Section II, the couplingand trade-off effects within the hierarchical control structureare analyzed. In Section III, containment and consensus-baseddistributed coordination control strategy is introduced in detail.In Section IV, the small-signal model and its stability analysisare provided. In Section V, experimental results are presentedto prove the effectiveness of proposed controller. Finally, thepaper is concluded in Section VI.

II. COUPLING AND TRADE-OFF ANALYSIS WITHIN THE

HIERARCHICAL CONTROL

This section gives the coupling analysis among Q–V droopgains, line impedance differences, reactive power sharing er-rors, and relative deviations of voltage magnitudes. Then, thetrade-off effects between the accurate reactive power sharingregulation and voltages regulation are analyzed.

A. Coupling Analysis in the Primary Control Level

The simplified islanded MG for analysis purposes is shownin Fig. 1, including two DGs, two local loads, and one commonload.

During the islanded operation, DG units are operated underconventional Q–V droop control, defined as

EDGi = E∗ − niQi (1)

where EDGi is the reference voltage for the inner control loop,E∗ is the voltage magnitude reference of droop control, ni isthe Q–V droop gain, and Qi is the output reactive power.

HAN et al.: CONTAINMENT AND CONSENSUS-BASED DISTRIBUTED COORDINATION CONTROL TO ACHIEVE BOUNDED VOLTAGE 5189

Fig. 2. Coupling analysis results.

In Fig. 1, according to the line impedances, the voltage dropcan be calculated as [24]

EDGi − Vpcc =Ri (Pi − PLi) + Xi (Qi − QLi)

V ∗ (2)

where V ∗ is the nominal voltage according to the system re-quirement, Ri and Xi are the line impedance, Pi and Qi arethe output active and reactive powers, and PLi and QLi are thelocal active and reactive powers for ith DG.

From (1) and (2), the coupling effects of droop gains andline impedance ratios on reactive power sharing error and volt-age magnitude deviations can be analyzed by using the controlvariate method, shown in Fig. 2. For convenience, same droopgains, n1 = n2 = n, are assumed in this analysis. Note that inthis figure logarithm horizontal x-axis is used to show clearlythe trends.

Fig. 2(a-1) shows the effect of absolute Q–V droop gains andthe ratio of line impedances on reactive power sharing error.To be more specific, Fig. 2(a-2) shows the relationship betweenabsolute Q–V droop gains and reactive power sharing error.It is shown that the reactive power sharing error can be ef-fectively reduced by increasing droop gains regardless of lineimpedance ratios. Fig. 2(a-3) shows the relationship betweenthe line impedance ratios and the reactive power sharing error.It is shown that with the increasing of line impedance ratios, thereactive power sharing error is increased, while larger absoluteQ–V droop gain can reduce this error.

From the other standpoint, Fig. 2(b-1) shows the effect ofabsolute Q–V droop gains and the line impedance ratios onrelative deviations of output voltage magnitudes of DG units.As shown in Fig. 2(b-2), both smaller and larger droop gainscan help to reduce the relative deviations of voltage magnitudes.

Here, it is emphasized that due to the feature of Q–V droopcontrol, the voltage magnitudes are always deviated from thenominal value and in this section, the relative deviations betweenvoltage magnitudes of only two DGs are discussed to make theexplanation easier. The voltage restoration will be discussedin Section II-B. In addition, there exist several peak values fordifferent line impedances conditions shown in Fig. 2(b-2), whichis also proven by Fig. 2(b-3). Thus, by combining the results withthose from Fig. 2(a-1)–(a-3), the relatively larger droop gainswithin the stability range can relieve the reactive power sharingerror and voltage magnitude relative deviations effectively andsimultaneously.

Based on the above-mentioned analyses, the following fourmain conclusions can be obtained.

1) Both larger and smaller droop gains can help decreasingthe relative deviations of output voltage magnitudes toreduce the reactive circuit current.

2) Larger droop gains can weaken the effect of lineimpedance differences and decrease the reactive powersharing errors.

3) Based on 1) and 2), larger droop gains are more suitable todecrease two errors under the assumption that the droopgain can satisfy the stability requirements.

4) The absolute droop gains and line impedances have de-cisive influence over the relative voltage deviations andreactive power sharing errors rather than the relativeratios.

B. Trade-Off Analysis Between Voltage Magnitude Regulationand Reactive Power Sharing in the Secondary Control Level

Due to the features of above-mentioned analyses in the pri-mary level, two control objectives should be achieved: 1) volt-

5190 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 53, NO. 6, NOVEMBER/DECEMBER 2017

Fig. 3. Voltage magnitudes regulation control for two parallel DGs withsmaller droop gain and larger line impedance difference.

Fig. 4. Voltage magnitudes regulation control for two parallel DGs with largerdroop gain and smaller line impedance difference.

Fig. 5. Reactive power sharing regulation control for two parallel DGs withsmaller droop gain and larger line impedance differences.

age recovery; and 2) accurate reactive power sharing in thesecondary level. In this section, it is analyzed that trade-off ef-fects always exist between the two objectives with different lineimpedances from DG units.

To analyze the trade-off effects between voltage regulationand reactive power sharing regulation under different conditions,Figs. 3–6 are depicted, in which black lines denote the conditionbefore secondary control regulation and red lines stand for thecondition after secondary control regulation. The analyses aredivided in two parts. Figs. 3 and 4 show only the effect of voltageregulation, while Figs. 5 and 6 show only the effect of reactivepower sharing regulation.

Fig. 6. Reactive power sharing regulation control for two parallel DGs withlarger droop gain and smaller line impedance differences.

Fig. 3 shows the condition with smaller Q–V droop gain andlarger line impedance difference, after regulating the voltagemagnitude to the nominal value, Q1 becomes much smallerand Q2 becomes much larger than before. If larger Q–V droopgain and smaller line impedance difference are considered, asshown in Fig. 4, after voltage regulation, the deviation betweenQ1 and Q2 also becomes larger. Thus, no matter what kind ofparameter conditions are chosen in the system, the trade-offeffect is observable.

In Fig. 5, a smaller Q–V droop gain and a larger lineimpedance difference condition are assumed, after reactivepower sharing regulation, the voltage deviation between twoDGs is enlarged. In Fig. 6, a relative larger Q–V droop gain anda smaller line impedance difference are assumed as another con-dition, after the reactive power sharing regulation, the voltagedeviation from two DGs is almost not changed.

Thus, regulating voltage magnitudes to nominal value cancause larger deviation of reactive power sharing. By comparison,reactive power sharing regulation can cause relative little effectson voltage magnitude deviations. Based on above-mentioneddiscussions, a reasonable operation strategy is to ensure accuratereactive power sharing regardless of relative voltage deviationsbetween buses, but necessarily, bounding voltage magnitudeswithin a reasonable range defined by standards to guarantee thestability and power quality.

III. PROPOSED DISTRIBUTED COORDINATION CONTROL FOR

BOUNDING VOLTAGE DEVIATIONS AND REACTIVE

POWER SHARING

This section explains the proposed distributed coordinationcontrol in detail. Comparing with the algorithms presented in theliterature [14], [17], [19], the proposed algorithm can achieve thecontrollability of all the bus voltages instead of only controllingthe average value of them. A containment-based voltage con-troller is first proposed to bound all the bus voltages into allowedrange, which can guarantee the power quality of the system evenunder worst conditions, especially for the system with the largerline impedance differences or the system with necessary plug-and-play operations. Compared with the proposed method, theconventional average value based controller may result in over-/under-voltage ranges in some of the buses although the averagevalue of all bus voltages can be kept at nominal value. At thesame time, a consensus-based reactive power controller is also

HAN et al.: CONTAINMENT AND CONSENSUS-BASED DISTRIBUTED COORDINATION CONTROL TO ACHIEVE BOUNDED VOLTAGE 5191

Fig. 7. Configuration of the containment-based and consensus-based distributed coordination controller.

involved in the proposed control algorithm to achieve properreactive power sharing. Thus, the proposed two controllers canachieve the coordinated control including the global boundedbus voltages and reactive power sharing simultaneously. A hi-erarchical control structure can be formulated to integrate ofmultiple functions seamlessly.

A. Definitions and Notations

For the control system with n distributed controllers, a con-troller is called leader if it only provides information to itscommunication neighbors and does not receive information. Acontroller is called follower if it can receive information fromone or more neighbors through communication topology. Let Ni

denote the set of ith-controller neighbors chosen from followers,and Ri denote the set of leaders, which can send its informationto ith-agent directly. The definition mentioned above is appliedto containment-based voltage controller. At the same time, theconsensus-based reactive power controller only uses the neigh-bors’ information without the leaders’ information.

Let C be a set in a real vector space V ⊆ Rp . The set C is calledconvex if, for any x and y in C, the point (1 − z)x + zy is alsoin C for any z ∈ [0, 1]. The convex hull for a set of points X ={x1, . . . , xq} in V is the minimal convex set containing all pointsin X. Let Co(X) denote the convex hull of X. In particular, whenV ⊆ R, Co(X) = {x|x ∈ [min xi,max xi ]}which will be usedin following. In addition, define vector Z ∈ Rn , then diag(Z) ∈Rn×n as the diagonal matrix whose diagonal elements are theelements in vector Z. In is the unit matrix and 0n is the zeron × n matrix.

For consensus-based control, an adjacency matrix is definedas A = [aij ] ∈ Rn×n with aij > 0 if node i can receive in-formation from node j otherwise aij = 0. The Laplacian ma-trix is defined as LQ = [lij ] ∈ Rn×n with lii =

∑nj=1 aij and

lij = −aij , i �= j.For containment-based control, the range is formed by two

leaders which are called the lower and upper voltage boundaries.Thus, another adjacency matrix is defined as B = [bij ] ∈ Rn×2

with bil = 1 if node i can receive information from one of thetwo leaders otherwise bil = 0, in which l represents the label of

two leaders; another Laplacian matrix is defined as LE = [lij ] ∈Rn×(n+2) with lii =

∑nj=1 aij +

∑n+2l=n+1 bil for other rows,

when j < n, lij = −aij , otherwise when j > n, lij = −bij .

B. Proposed Coordination Controller

The containment-based controller generates a correction termeEi for each DG to keep the voltage within a range which is asconvex hull. The controller expression is defined as

eE i = −∑

j∈Ni

aij (EDGi − EDGj ) −∑

l∈Ri

bil (EDGi − El) (3)

where EDGi and EDGj are the voltage magnitudes of ith DG andjth DG, respectively, and El is the voltage leader which can beeither upper boundary EU bou or lower boundary ELbou.

Equation (3) can be written into matrix form as

eE = −LE E (4)

where EDG = [EDG1, . . . EDGn ]T , Eleader = [EU bou ELbou ]T ,E = [ET

DG ETleader ]

T , and eE = [ eE 1 . . . eEn ]T .Then the error eE is fed into a proportional-integral (PI)

controller.Consensus-based reactive power controller is defined as

enQi = −∑

j∈Ni

aij (niQi − njQj ) (5)

where ni and nj are the reactive power droop gains, and Qi andQj are the output reactive for ith DG and jth DG.

Equation (5) can be written into matrix form as

enQ = −LQNQ (6)

where N = diag{[n1, . . . nn ]T } and enQ = [ enQ1 . . . enQn

]T .Then, the error enQ is fed into another PI controller.The configuration of the proposed controller is shown in

Fig. 7, including the containment-based voltage controller andthe consensus-based reactive power controller. The informationformat from DGs (followers) is defined as Υf j = [njQj , Vj ],and the information format from the leader is defined asΥl = [ 0, El ].

5192 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 53, NO. 6, NOVEMBER/DECEMBER 2017

C. Different Work Conditions

The proposed algorithms are based on the hierarchical struc-ture, which consists of primary, secondary, and tertiary levels.In the primary level, the stability can be guaranteed by the con-ventional droop controller. The proposed two controllers areimplemented in the secondary level, corresponding to two crit-ical operation conditions.

Condition 1: Under this condition, DGs have enough powercapacity to support voltage and reactive power regulations. Thecontainment-based controller guarantees that all the bus voltagesare kept within a prescribed range defined by standards, whileconsensus-based controller realizes proportional reactive powersharing by all the DGs at the same time.

Condition 2: Under this condition, all the bus voltages arecontrolled to be maintained within a dynamic boundary whichcould be adapted based on the command from the higher controllevel. Even though the voltage boundary may change, the totalreactive power is proportional shared among DGs. It is worthsaying that the principle about how to change the voltage bound-ary is out of the scope of this paper, which should be designedin the tertiary level to achieve power management operation.

Notice that, under both work conditions, the two controllersare activated at the same time. The bounded voltage deviationsand precise reactive power sharing can be achieved simultane-ously. Both the work conditions have been tested in Section V-A.

IV. SMALL-SIGNAL STABILITY ANALYSIS

This section develops the small-signal model for stabilityanalysis and parameters design purposes. The model includesthe proposed containment-based voltage controller, consensus-based reactive power controller, active and reactive power cal-culation, low-pass filter, and droop controller. The whole modelis based on the synchronous reference frame.

A. Small-Signal Model for Proposed Controllers

For the containment-based voltage controller shown in (4),the small signal model is expressed as

ΔeE = −L′E ΔEDG (7)

where L′E is the matrix which deletes the last two columns

of matrix LE neglecting the dynamic of leaders ΔeE =[ΔeE 1 . . . ΔeEn ]T and ΔEDG = [ΔEDG1 . . . ΔEDGn ]T .

For the consensus-based reactive power controller in (6), thesmall signal model is expressed as

ΔenQ = −LQNΔQ (8)

where ΔenQ = [ΔenQ1 . . . ΔenQn ]T and ΔQ = [ΔQ1 . . .ΔQ]T .

Considering the dynamic of voltage changes, by adding avoltage disturbance term EDG in left part of (1), then it can berewritten as

EDG = E∗ − EDG − NQ (9)

which can be seen as the dynamic of the primary level control.

As explained above, the two proposed controllers should pro-vide control signals added into (9) through PI controllers. Thus,the system can be written as

ΔEDG = − ΔEDG − NΔQ − KpQLQNΔQ

− KpE L′E ΔEDG + KiQΔenQ + KiE ΔeE (10)

where KpQ = diag([kpQ1 . . . kpQn ]T ) corresponds to the pro-

portional parameters, KiQ = diag([kiQ1 . . . kiQn ]T ) to theintegral parameters of the PI controllers for the consensus-based reactive power controller, KpE = diag([kpE 1 . . .kpEn ]T ) to the proportional parameters, and KiE =diag([kiE 1 . . . kiEn ]T ) to the integral parameters of the PIcontrollers for the containment-based voltage controllers.

Due to the low-pass filter effect, the small-signal model ofoutput reactive power Qi can be written as

ΔQ = −ωcΔQ + ωcΔq (11)

where ωc is the cut-off frequency of low-pass filter and theinstant output reactive power is Δq = [Δq1 . . . Δq

n]T .

Considering synchronous reference frame for the ith DG, thevector voltage �EDGi can be written as

�EDGi = Edi + jEqi (12)

where Edi =EDGi cos δi, Eqi =EDGi cos δi, and δi = arctan(Edi/Eqi).

Linearizing (12) of δi , we can get

Δδi = (∂δi/∂Edi) ΔEdi + (∂δi/∂Eqi) ΔEqi

= mdiΔEdi + mqiΔEqi (13)

where mdi = −Eqi/(E2di + E2

qi) and mqi = Edi/(E2di + E2

qi).Since Δωi(s) = sΔδi(s), (13) can be rewritten as

Δωi = mdiΔEdi + mqiΔEqi . (14)

Considering that EDGi = | �EDGi | =√

E2di + E2

qi , it can be

linearized as

ΔEDGi = ndiΔEdi + nqiΔEqi (15)

where ndi = Edi/√

E2di + E2

qi and nqi = Eqi/√

E2di + E2

qi .

It follows:

ΔEDGi = ndiΔEdi + nqiΔEqi . (16)

Thus, from the equation set consisting of (14) and (16) forvariables ΔEdi and ΔEqi , we have

{ΔEdi = m1iΔω + m2iΔEDGi

ΔEqi = m3iΔω + m4iΔEDGi

(17)

where m1i = nqi/(mdinqi − mqindi), m2i = −mqi/(mdinqi − mqindi), m3i = ndi/(mqindi − mdinqi), andm4i = −mdi/(mqindi − mdinqi).

HAN et al.: CONTAINMENT AND CONSENSUS-BASED DISTRIBUTED COORDINATION CONTROL TO ACHIEVE BOUNDED VOLTAGE 5193

Substituting (10) and (15) into (17) and writing into matrixform as⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

ΔEd = M1Δω + A1NdΔEd + A1NqΔEq

+A2ΔQ + M2KiE ΔeE + M2KiQΔenQ

ΔEq = M3Δω + B1NdΔEd + B1NqΔEq

+B2ΔQ + M4KiE ΔeE + M4KiQΔenQ

(18)

where M1 =diag([m11 . . . m1n ]T ), M2 =diag([m21 . . . m2n ]T ),M3 = diag([m31 . . . m

3n]T ), M4 = diag([m41 . . . m

4n]T ),

Nd =diag([nd1 . . . ndn

]T ), Nq =diag([nq1 . . . nqn

]T ), A1 =−M2(In + KpE L′

E ), A2 =−M2(In + KpQLQ )N , B1 =−M4

(In + KpE L′E ), B2 = −M4(In + KpQLQ )N , ΔEd = [ΔEd1

. . . ΔEdn ]T , ΔEq = [ΔEq1 . . . ΔEqn ]T , and Δω = [Δω1

. . . Δωn ]T .In addition, considering the active power droop control and

the low-pass filter effect

Δω = −ωcΔω − ωcMΔp (19)

where M = diag([m1 . . . mn ]T ) is the P-f droop gain andΔp = [Δpi . . . Δpn ] is instant active power.

B. Small-Signal Model for the Whole System

Considering load and line impedances together, the conduc-tance matrix G and susceptance matrix B can be written as

G =

⎡

⎢⎢⎣

G11 · · · G1n

.... . .

...

Gn1 · · · Gnn

⎤

⎥⎥⎦ , B =

⎡

⎢⎢⎣

B11 · · · B1n

.... . .

...

Bn1 · · · Bnn

⎤

⎥⎥⎦ . (20)

Based on the KCL and KVL theorems, the small-signal modelrepresenting the relationship between output current and voltagecan be written as

{ΔId = GΔEd + (−B) ΔEq

ΔIq = BΔEq + GΔEd

(21)

where ΔId = [ΔId1 . . . ΔIdn ]T and ΔIq = [ΔIq1 . . . ΔIqn ]T .Since instant active and reactive powers are obtained through

an orthogonal system as{

pi = 3/2 (EdiIdi + EqiIqi)

qi = 3/2 (EqiIdi − EdiIqi). (22)

The small-signal model of the instant output power is pre-sented as{

Δp = 3/2 (IdΔEd + IqΔEd + EdΔId + EqΔIq )

Δq = 3/2 (−IqΔEd + IdΔEd + EqΔId − EdΔIq )(23)

where Id = diag([Id1 . . . Idn ]T ), Iq = diag([Iq1 . . . Iqn ]T ),Ed = diag([Ed1 . . . Edn ]T ), and Eq = diag([Eq1 . . . Eqn ]T ).

By combining (21) and (23), the small signal model of theinstant active and reactive powers can be expressed as

{Δp = S1ΔEd + S2ΔEq

Δq = S3ΔEd + S4ΔEq

(24)

where S1 = 3/2(Id + EdG + EqB), S2 = 3/2(Iq − EdB +EqG), S3 = 3/2(−Iq + EqG − EdB), and S4 = 3/2(Id −EqB − EdG).

By substituting (24) into (11) and (19), and by substituting(15) into (7) and combining with (8) and (18), we can obtain thewhole system model as follows:

X = FX (25)

where F =⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−ωcIn −ωcMS1 −ωcMS1 0n 0n 0n

M1 A1Nd A1Nq A2 M2KiE M2KiQ

M3 B1Nd B1Nq B2 M4KiE M4KiQ

0n ωcS3 ωcS4 −ωcIn 0n 0n

0n −L′E Nd −L′

E Nq 0n 0n 0n

0n 0n 0n −LQN 0n 0n

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

and X = [ΔωT ΔETd ΔET

q ΔQT ΔeTE ΔeT

nQ ]T .In order to make the modeling process more clear, Fig. 8

shows the equivalent small signal model diagram for the pro-posed controller and the ac MG system.

C. Stability Analysis

In order to analyze the model quantitatively, an MG includingfour parallel connected DGs, a local load, and a common loadare considered as a study case. In this section, root locus plotsare shown to reflect the dynamical behavior of the system byconsidering different control parameters.

Fig. 9 shows the root locus movement considering the pro-portional coefficient KpE of PI controller for containment-basedcontrol changing from 7 to 15. From the enlarged part in Fig. 9,it is shown that the two dominating poles located near the imag-inary axis are moving toward the real axis and away from theimaginary axis, which indicate that the system is becomingmore damped. Six complex poles which are also affected byKpE are moving away from the imaginary axis, thus improvingthe response speed.

Fig. 10 shows root locus considering integral coefficient KiE

of the PI controller for containment-based control changingfrom 1 to 100. From the enlarged part in Fig. 10, it is shown thattwo dominating poles are moving away from the real axis, whichmeans that the system is becoming less damped. Simultaneously,two poles on the real axis are moving away from original point.The rest six complex poles are less affected than the proportionalcoefficient KpE .

Fig. 11(a) shows root locus considering proportional coef-ficients KpQ for a PI controller for consensus-based controlchanging from 7 to 15. It can be observed that two dominat-ing poles in the blue circle are barely affected. In addition, one

5194 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 53, NO. 6, NOVEMBER/DECEMBER 2017

Fig. 8. Small signal model diagram for the whole system.

Fig. 9. Root locus plot 7 < KpE < 15.

Fig. 10. Root locus plot 1 < KiE < 100.

pole on the real axis moves toward origin point, which canslow down the response speed. Six complex poles are mov-ing away from real axis, which means the system is becomingless damped. Fig. 11(b) shows root locus considering integral

Fig. 11. Root locus: (a) 7 < KpQ < 15; and (b) 30 < KiQ < 120.

Fig. 12. Root locus: (a) common load from (15 + 15.7j) to (1500 +1570j) Ω; and (b) local load from (30 + 31.4) to (3000 + 3140j) Ω.

TABLE ICONCLUSION FROM STABILITY ANALYSIS

Containment-based controller Consensus-based controller

↑ KpE Response speed ↑ ↑ KpQ Response speed ↓Damping ↑ Damping ↓

↑ KiE Response speed – ↑ KiQ Response speed ↑Damping ↓ Damping ↓

coefficients KiQ of PI controller for consensus-based controlchanging from 30 to 120. The two dominating poles in the bluecircle are also not affected. One dominating pole on the real axis

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TABLE IIELECTRICAL DATA COMPARISON

Category In this paper In [ 17] In [ 19]

Line Impedance Z12 = 1.2 Ω + 5.4 mH;Z23 = 0.4 Ω + 1.8 mH;Z34 = 0.8 Ω + 3.6 mH

Z12 = 0.8 Ω + 3.6 mH;Z23 = 0.4 Ω + 1.8 mH;Z34 = 0.7 Ω + 1.9 mH

Z12 = 0.8 Ω + 3.6 mH;Z23 = 0.4 Ω + 1.8 mH;Z34 = 0.7 Ω + 1.5 mH

Nominal Voltage 325 V 325 V 120 VTotal Reactive Power 3000 W 1260 W 750 W

Fig. 13. Root locus: (a) Changing from 0.8∗ Z12 to 1.2∗ Z12; (b) changingfrom 0.8∗ Z23 to 1.2∗ Z23; and (c) changing from 0.8∗ Z34 to 1.2∗ Z34.

is moving away from the original point, which can increase thesystem response speed. Six complex poles are moving towardthe imaginary axis which makes the system be less damped.

Fig. 12(a) and (b) shows that the eigenvalues are not affectedby load changes, including common and local loads, indicatinggood robustness of the proposed controllers. Thus, the proposedscheme does not require prior knowledge of the load informationin the system. To be clearer, the whole analysis conclusion issummarized in Table I.

The analyses shown in Figs. 9 –12 are based on the completesystem information including the line impedance values whichare difficult to know in real MG system. The root locus plots bychanging the line impedance values from 0.8 to 1.2 times of thereal values are shown in Fig. 13. It is shown that by changingthe line impedance values cannot affect the poles much thanthat from the control parameters. The result can also be derivedby comparing the ranges of imaginary and real axis shown inFig. 13, and that in Figs. 9–12. Thus, the parameter analysismethod and results can be used for real MG system designwithout knowing the accurate value of line impedances. The

Fig. 14. Experimental setup in the AAU-Microgrid Research Laboratory.

values of the line impedances for this analysis are presented inTable II.

V. EXPERIMENTAL RESULTS

The proposed control scheme is implemented and tested in anexperimental MG setup operating in islanded mode, shown inFig. 14 at the AAU-Microgrid Research Laboratory. The setupconsists of four parallel-configured power electronics inverters,a real-time control and monitor platform, LCL filters, and RLloads. Communication link is only built between neighboringunits shown in the top left corner of Fig. 13. Converters ratedactive and reactive power are 2:2:1:1 for DG1 − DG4. The nom-inal voltage magnitude is set to 325 V with 1% voltage boundary(325 ± 1%). The experimental results are shown in Figs. 15–19.At t = T0, four DGs are controlled by the conventional droopcontrol, and at t =T1 the proposed controller is enabled.

A. Case 1: Performance Assessment and Comparison

Fig. 15 shows the performance comparison between the con-ventional droop controller and the proposed one. Fig. 15(a)shows the voltage performance and Fig. 15(b) shows the reac-tive power sharing characteristic. Before t = T1, the system iscontrolled by the conventional droop controller. The bus volt-age magnitudes are dropped more than 18 V, which exceedthe lower voltage boundary 320 V. The reactive power sharingamong four DGs does not follow the proportionality 2:2:1:1. Att = T1, the proposed controller is activated. Then, the bus volt-

5196 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 53, NO. 6, NOVEMBER/DECEMBER 2017

Fig. 15. Performance evaluation of the proposed controller. (a) Performanceof output voltage bound. (b) Performance of reactive power sharing.

age magnitudes can be bounded within prescribed range and thereactive power is proportionally shared to 2:2:1:1. Furthermore,between t = T2 and t = T4, the boundary is changed and thebus voltage magnitudes are followed the changed boundary intothe new range which is between 320 and 310 V. In addition, theperformance of reactive power sharing can also be guaranteed,being still equal to 2:2:1:1.

In addition, at t = T3, the load is changed and the perfor-mance of reactive power sharing is also kept accurate. Aftert = T4, the voltage boundary is changed back to the nominalrange and both the voltage and reactive power sharing perfor-mance are kept well. It is shown that after activating the proposedcontroller, the bus voltage magnitudes can be bounded withinthe dynamic range. Meanwhile the output reactive power can beproportionally shared to 2:2:1:1 during the whole process.

B. Case 2: Communication Failure Resiliency

Resiliency to a single communication link failure is studiedin Fig. 16.

Fig. 16. Resiliency to communication failure between DG2 and DG3. (a) Per-formance of output voltage bound. (b) Performance of reactive power sharing.

The communication link between DG2 and DG3 has been dis-abled at t = T6. As shown in the zoomed in part of Fig. 16(a)and (b), after small oscillations occur around ±0.5 V, it doesnot have any impact on the performance of bounded bus voltageand reactive power sharing. After that, the load is switched att = T7 and T8. The performance is also kept well. It is con-cluded that both the dynamic and steady-state performance ofthe proposed controller cannot be affected as long as the com-munication network remains connected from the perspective ofgraph theory.

C. Case 3: Plug-and-Play Comparison Study

Fig. 17 shows the plug-and-play capability of the proposedcontroller. DG4 is unplugged at t = T9. Thus, the bus volt-age and reactive power from DG4 decay to zero. Notice thata source failure also means loss of communication links con-nected to other DGs. As shown in Fig. 17(a), the bus voltagesof DG1- DG3 are kept inside the acceptable range. In addition,Fig. 17(b) shows the per-unit value of output reactive power todecrease the range of y-axis, indicating that the per-unit val-ues of output reactive power among DG1- DG3 are all equal to

HAN et al.: CONTAINMENT AND CONSENSUS-BASED DISTRIBUTED COORDINATION CONTROL TO ACHIEVE BOUNDED VOLTAGE 5197

Fig. 17. Plug-and-play study for DG4 under proposed controller. (a) Perfor-mance of output voltage bound. (b) Performance of reactive power sharing.

Fig. 18. Plug-and-play study for DG 4 under controller in [17]. (a) Perfor-mance of output voltage bound. (b) Performance of reactive power sharing.

Fig. 19. Plug-and-play study for DG 4 under controller in [19]. (a) Perfor-mance of output voltage bound. (b) Performance of reactive power sharing.(c) Average voltage value.

0.55 p.u. At t = T10, DG4 begins to synchronize the frequencyand phase with the MG. After successful synchronization, att = T11, DG4 is connected without activating the proposedcontroller. At t = T12, the proposed controller and commu-nication are activated for DG4. Fig. 17 shows that bus voltagemagnitudes can be bounded within the range and the per-unitvalue of reactive power sharing among four DGs are equal to0.47 p.u. after activating the proposed controller for DG4.

Furthermore, to make the proposed containment-based volt-age controller more convincing, the control performance by

5198 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 53, NO. 6, NOVEMBER/DECEMBER 2017

using controllers of [17] and [19], which are very popular in theliterature about voltage and reactive power sharing regulation, isshown in Figs. 18 and 19 by using the same electrical topology,respectively.

Fig. 18 shows the plug-and-play performance of the controllerproposed in [17]. When DG4 is disconnected from the MG, thebus voltage of DG2 exceeds the voltage boundary which canaffect the power quality of the system. In addition, as shownin Fig. 19, by using the controller proposed in [19], when DG4

is plugged out of the system, even though the average voltagevalue can be kept at 325 V, shown in Fig. 19(c), the bus voltageof DG2 also exceeds the voltage boundary shown in Fig. 19(a).

Notice that the plug-and-play tests are also shown in [17] and[19] without observing this problem. The reason is that by com-paring the electrical parameters of this paper with the ones in[17] and [19], the line impedance deviations that we use in thispaper are much larger, and thus the total reactive power as well.The electrical data comparison is shown in Table II in detail.Thus, it is proved that the proposed containment-based con-troller is more suitable for larger systems with relatively largerline impedance deviations and higher reactive power sharingrequirements.

Remark 1: If the line impedance deviations in the system aremuch higher and some DGs are disconnected of the system, it isnot possible to guarantee both the performance of bus voltagesbound and reactive power sharing simultaneously due to theelectrical limitations. Under these conditions, the advantage ofproposed controller is that it can decide either to guarantee thebus voltage bound through setting error saturation of reactivepower sharing performance, or to guarantee the reactive powersharing performance through enlarging the voltage boundaryof the system according to the MGs or electrical distributionstandards. Previously proposed controllers cannot perform thishigh level of freedom to achieve global bus voltage bound, sothat the control performance of all global voltages in the systemcannot be compromised among DGs, especially under extremeconditions.

VI. CONCLUSION

This paper presents the coupling/trade-off effects betweenvoltage regulation and reactive power sharing in different levelsof a hierarchical control structure. The coupling effects amongQ–V droop gains, line impedance differences, reactive powersharing errors, and voltage magnitudes deviations are analyzedin the primary control level. The trade-off effects between theaccurate reactive power sharing and voltage magnitudes reg-ulations are further analyzed in the secondary level. A fullydistributed coordination controller including both containment-based and consensus-based controllers is proposed to offer ahighly flexible and reliable operation of DGs, thus achievingboth bound the voltage magnitudes within a reasonable rangeand achieving accurate reactive power sharing. A detailed smallsignal model is derived and the effects of different parame-ters change for the proposed controller are analyzed. Experi-mental results are presented to demonstrate the effectivenessof proposed controller including performance assessment and

comparison, resiliency for communication failure, and plug-and-play study.

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Renke Han (S’16) received the B.S. degree in au-tomation, and the M.S. degree in control theory andcontrol engineering from Northeastern University,Shenyang, China, in 2013 and 2015, respectively. Heis currently working toward the Ph.D. degree in powerelectronics system with the Department of EnergyTechnology, Aalborg University, Aalborg, Denmark.

Currently, he is a guest Ph.D. student withthe Laboratoire d’Automatique, Ecole polytechniquefederale de Lausanne, Lausanne, Switzerland, underthe supervision of Prof. Giancarlo Ferrari-Trecate.

His research interests include the distributed control, event-triggered control,and plug-and-play control to achieve stability operation, and reasonable powersharing for ac and dc microgrid.

Mr. Han received the Outstanding Presentation Award from the AnnualConference of the IEEE Industrial Electronics Society, Italy, in 2016, and theOutstanding Master Degree Thesis Award from Liaoning Province, China, in2014.

Lexuan Meng (S’13–M’15) received the B.S. de-gree in electrical engineering, and the M.S. degreein electrical machine and apparatus from the NanjingUniversity of Aeronautics and Astronautics, Nanjing,China, in 2009 and 2012, respectively, and the Ph.D.degree in power electronic systems from the Depart-ment of Energy Technology, Aalborg University, Aal-borg, Denmark, in 2015.

He is currently a Post-Doctoral Researcher withthe Department of Energy Technology, Aalborg Uni-versity, working on flywheel energy storage and on-

board electric power systems. His research interests include microgrids, gridintegration of energy storage systems, power quality, and distributed control.

Giancarlo Ferrari-Trecate (SM’12) received thePh.D. degree in electronic and computer engineeringfrom the University of Pavia, Pavia, Italy, in 1999.

Since September 2016, he has been a Professorwith the Ecole polytechnique federale de Lausanne,Lausanne, Switzerland. From 1998 to 2001, he wasas a Postdoctoral Fellow at ETH Zurich, Zurich,Switzerland. In 2002, he joined the French Institutefor Research in Computer Science and Automation,Rennes, France, as a Research Fellow. From Marchto October 2005, he was at the Politecnico di Milano,

Milan, Italy. From 2005 to 2016, he was an Associate Professor in the Universityof Pavia. His research interests include scalable control, control of microgrids,analysis of biochemical networks, hybrid systems, and Bayesian learning.

Dr. Ferrari-Trecate received the Researcher Mobility Grant from theItalian Ministry of Education, University and Research, in 2005. He is cur-rently a member of the International Federation of Automatic Control (IFAC)Technical Committee on Control Design and is on the editorial board of Auto-matica and Nonlinear Analysis: Hybrid Systems.

Ernane Antonio Alves Coelho (M’12) received theB.S. degree in electrical engineering from the Fed-eral University of Minas Gerais (UFMG), Belo Hor-izonte, Brazil; the M.S. degree in power electron-ics from the Federal University of Santa Catarina,Florianopolis, Brazil; and the Ph.D. degree in powerelectronics from UFMG, in 1987, 1989, and 2000,respectively.

In 1989, he joined the Electrical EngineeringFaculty, Federal University of Uberlandia (UFU),Uberlandia, Brazil, where he is currently a Full Pro-

fessor in the Power Electronics Research Group. In 2014, he was a VisitingProfessor in the Microgrid Research Group, Department of Energy Technology,Aalborg University, Aalborg, Denmark. His research interests include power-factor correction, photovoltaic and fuel cell systems, microgrid modeling, anddigital control by microcontrollers and digital signal processors.

Juan C. Vasquez (M’12–SM’14) received theB.S. degree in electronics engineering from theAutonomous University of Manizales, Manizales,Colombia, and the Ph.D. degree in automatic con-trol, robotics, and computer vision from the Tech-nical University of Catalonia, Barcelona, Spain, in2004 and 2009, respectively.

In 2011, he was an Assistant Professor and since2014 he has been an Associate Professor with the De-partment of Energy Technology, Aalborg University,Aalborg, Denmark, where he is the Vice Programme

Leader of the Microgrids Research Program. His current research interestsinclude operation, advanced hierarchical and cooperative control, and the inte-gration of Internet of Things into the SmartGrid.

Dr. Vasquez is an Associate Editor of IET Power Electronics. He is cur-rently a member of the IEC System Evaluation Group SEG4 on Low VoltageDC Distribution and Safety for Use in Developed and Developing Economies,the Renewable Energy Systems Technical Committee TC-RES, and the IEEEIndustrial Electronics, Power Electronics, Industry Applications, and Power andEnergy Societies.

Josep M. Guerrero (S’01–M’04–SM’08–F’15) re-ceived the B.S. degree in telecommunications engi-neering, the M.S. degree in electronics engineering,and the Ph.D. degree in power electronics from theTechnical University of Catalonia, Barcelona, Spain,in 1997, 2000, and 2003, respectively.

Since 2011, he has been a Full Profes-sor with the Department of Energy Technology,Aalborg University, Aalborg, Denmark, where heis responsible for the Microgrid Research Program.Since 2012, he has been a Guest Professor with the

Chinese Academy of Science, Beijing, China, and the Nanjing University ofAeronautics and Astronautics, Nanjing, China. Since 2014, he has been a ChairProfessor with Shandong University, Jinan, China. Since 2015, he has been aDistinguished Guest Professor with Hunan University, Changsha, China. Since2016, he has been a Visiting Professor Fellow with Aston University, Birm-ingham, U.K., and a Guest Professor with Nanjing University of Posts andTelecommunications, Nanjing. His research interests encompass different mi-crogrid aspects, including power electronics, distributed energy-storage sys-tems, hierarchical and cooperative control, energy management systems, smartmetering, and the Internet of Things for ac/dc microgrid clusters and islandedminigrids; recently especially focused on maritime microgrids for electricalships, vessels, ferries, and seaports.

Dr. Guerrero is an Associate Editor for the IEEE TRANSACTIONS ON POWER

ELECTRONICS, the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, andthe IEEE INDUSTRIAL ELECTRONICS MAGAZINE, and an Editor for the IEEETRANSACTIONS ON SMART GRID and the IEEE TRANSACTIONS ON ENERGY

CONVERSION. He has been a Guest Editor of the IEEE TRANSACTIONS ON

POWER ELECTRONICS Special Issues on Power Electronics for Wind EnergyConversion and Power Electronics for Microgrids; the IEEE TRANSACTIONS ON

INDUSTRIAL ELECTRONICS Special Sections on Uninterruptible Power SuppliesSystems, Renewable Energy Systems, Distributed Generation and Microgrids,and Industrial Applications and Implementation Issues of the Kalman Filter; theIEEE TRANSACTIONS ON SMART GRID Special Issues on Smart DC DistributionSystems and Power Quality in Smart Grids; and the IEEE TRANSACTIONS ON

ENERGY CONVERSION Special Issue on Energy Conversion in Next-GenerationElectric Ships. He was the Chair of the Renewable Energy Systems TechnicalCommittee of the IEEE Industrial Electronics Society. He received the Best Pa-per Award from the IEEE Transactions on Energy Conversion for 2014–2015,and the Best Paper Prize from the IEEE-Power and Energy Society in 2015.He also received the Best Paper Award from the Journal of Power Electronicsin 2016. In 2014, 2015, and 2016, he was awarded by Thomson Reuters as aHighly Cited Researcher.